System and method for determining amount of radioactive material to administer to a patient

ABSTRACT

A computerized system and method are provided for determining an optimum amount of radioactivity to administer to a patient, comprising: assuming an activity retention limit; utilizing the activity retention limit to determine a dose rate for a phantom category; utilizing the dose rate for the phantom category to determine the dose rate for a second phantom category; and utilizing the dose rate for a second phantom category to find information regarding the second phantom category. In other embodiments, a computerized system and method are provided for determining an optimum amount of radioactivity to administer to a patient, comprising: obtaining at least one image relating to anatomy of a particular patient; obtaining multiple images regarding radioactivity distribution over time in the particular patient; combining the radioactivity images with the anatomy images; running a Monte Carlo simulation to obtain dose image information; and using the dose image information to obtain BED and/or EUD information.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Divisional of U.S. patent application Ser. No. 12/514,853, filed May 14, 2009, which is a National Stage of International Application No. PCT/US2007/085400 filed Nov. 21, 2007, which claims priority to U.S. Provisional Application No. 60/860,315 filed Nov. 21, 2006 and U.S. Provisional Application No. 60/860,319 filed Nov. 21, 2006. All of the foregoing are incorporated by reference in their entireties.

This invention was made with Government support under NIH/NCI grant RO1CA116477 and NIH/NCI grant RO1CA116477 and DOE grant DE-FG02-05ER63967. The authors also acknowledge the U.S. Government has certain rights in this invention.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates a method for determining an amount of radioactive material to administer to a patient, according to one embodiment.

FIG. 2 illustrates different types of models that can be used in determining dose rates for patients.

FIGS. 3-4 illustrate examples of actual images of patients.

FIG. 5 illustrates the output of a 3D-RD calculation, according to one embodiment.

FIG. 6 lists example reference phantom parameter values.

FIG. 7 provides example 48 h whole-body activity retention values for different phantoms and F₄₈ values.

FIG. 8 illustrates DRC values for different F₄₈ values.

FIG. 9 depicts examples of administered activity limits for different phantoms and F₄₈ values as a function of T_(E) when T_(RB) is set to 20 h.

FIG. 10 depicts example corresponding results when T equals 10 h.

FIG. 11 depicts examples of absorbed dose to lungs as a function of T_(E) for different F₄₈ values.

FIG. 12 depicts example results for the different phantoms at F₄₈=0.9 when DRC=20 cGy/h.

FIG. 13 illustrates absorbed dose v. T_(E) curves for adult female phantom and for two different remainder body effective clearance half-lives.

FIG. 14 illustrates a method for a 3D-RD calculation, according to one embodiment.

FIG. 15A illustrates an example of computed tomography (CT) images.

FIG. 15B illustrates an example of positron emission tomography (PET) images.

FIG. 15C illustrates an example of kinetic parameter images.

FIGS. 16-20 illustrates the use of EUD and BED formulas, according to several embodiments.

FIGS. 21-22 illustrates various parameter values.

FIG. 23 illustrates an example of how an outer shell (with 2 hour half life) is separated from an inner sphere (with 4 hour half life).

FIG. 24 illustrates an example of the impact of dose distribution on EUD.

FIG. 25 illustrates an example of the impact of dose distribution on EUD.

FIGS. 26A-26D illustrate various image.

FIG. 27 depicts an example of the DVH of the absorbed dose distribution obtained with 3D-RD, according to one embodiment.

FIGS. 28 and 29 depict example results obtained with the radiobiological modeling capabilities of 3D-RD, according to several embodiments.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION Utilizing Activity Retention Limit to Obtain Dose Rate

FIG. 1 illustrates details on how an activity retention limit is utilized to obtain the dose rate, according to one embodiment. The resulting dose rate (also referred to as absorbed dose rate) can, for example, make it possible to account for patient-specific differences in determining dose rates for patients. By moving to a dose-rate instead of an activity-based limit on the treatment prescription it is possible to apply well-established and clinically validated constraints on therapy to situations that differ from the original studies used to define these limits. It is also possible to consider the impact of combined, external radiotherapy with targeted radionuclide therapy in determining the optimal amount of radioactivity to administer to a patient. Once a dose-rate has been obtained, it is easier to use many available data sets to assist the patient. The following data sets are examples of data sets that can be utilized: Press et al., A Phase I/II Trial of Iodine-131-Tositumomab (anti-CD20), Etoposide, Cyclophosphamide, and Autologous Stem Cell Transplantation for Relapsed B-Cell Lymphomas, Blood, Volume 96, Issue 9, Pages 2934-42 (2000); Emami et al., Tolerance of Normal Tissue to Therapeutic Irradiation, Int. J. Radiation. Oncol. Biol. Phys., Volume 1991, Issue 21, pages 109-122. Those of ordinary skill in the art will see that many other data sets can also be utilized.

Choose Activity Retention Limit

In 105, an activity retention limit is chosen or assumed. This activity retention limit can be assumed based on data sets, such as the Benua-Leeper studies or other studies that have determined the maximum tolerated dose in a patient population. The Benua-Leeper studies have proposed a dosimetry-based treatment planning approach to ¹³¹I thyroid cancer therapy. The Benua-Leeper approach was a result of the observation that repeated ¹³¹I treatment of metastatic thyroid carcinoma with sub-therapeutic doses often fails to cause tumor regression and can lead to loss of iodine-avidity in metastases. The Benua-Leeper study attempted to identify the largest administered ¹³¹I radioactivity that would be safe yet optimally therapeutic. Drawing upon patient studies, the Benua-Leeper study formulated constraints upon the administered activity. For example, the Benua-Leeper study determined that the blood absorbed dose should not exceed 200 rad. This was recognized to be a surrogate for red bone marrow absorbed dose and was intended to decrease the likelihood of severe marrow depression, the dose-limiting toxicity in radioiodine therapy of thyroid cancer. In addition, it was determined that whole-body retention at 48 h should not exceed 120 mCi. This was shown to prevent release of ¹³¹I-labeled protein into the circulation from damaged tumor. Furthermore, in the presence of diffuse lung metastases, it was determined that the 48 h whole-body retention should not exceed 80 mCi. This constraint helped avoid pneumonitis and pulmonary fibrosis. Thus, for example, referring to FIG. 1, in 105, an 80 mCi activity retention limit at 48 h can be assumed based on the Benua-Leeper studies. As shown below, the 48 hr retention limit is converted to a dose-rate limit that can then be used to adjust the 80 mCi to a value that would deliver the same dose-rate for different patient geometries (e.g., pediatric patients)

Determine Dose-Rate in Reference Phantom

In 110 of FIG. 1, utilizing the activity retention limit chosen in 105, S-factors for different phantoms are used to convert the activity retention limit to a dose-rate DRQ). In equation (1) below, a dose-rate DR(t) for lungs at time t in reference phantom P is found:

DR^(P)(t)−A _(LU)(t)·S _(LU←LU) ^(P) +A _(RB)(t)·S _(LU←RB) ^(P)   (1)

with:

$\begin{matrix} {{{A_{LU}(t)} = {\frac{A_{T} \cdot F_{T}}{^{- \pi}{LU}^{\cdot T}}^{- \lambda}{LU}^{\cdot T}}},} & (2) \\ {{{A_{RB}(t)} = {\frac{A_{T} \cdot \left( {1 - F_{T}} \right)}{^{- \lambda}{RB}^{\cdot T}}^{- \lambda}{RB}^{\cdot T}}},} & (3) \\ {{S_{{LU}\leftarrow{RB}}^{P} = {{S_{{LU}\leftarrow{TB}}^{P} \cdot \frac{M_{TB}^{P}}{M_{TB}^{P} - M_{LU}^{P}}} - {S_{{LU}\leftarrow{LU}}^{P} \cdot \frac{M_{LU}^{P}}{M_{TB}^{P} - M_{LU}^{P}}}}},} & (4) \end{matrix}$

A_(LU)(t) lung activity at time t,|

S_(LU←LU) ^(P) lung to lung ¹³¹I S-factor for reference phantom, P,

A_(RB)(t) remainder body activity (total-body-lung) at time, t,

S_(LU←RB) ^(P) remainder body to hung ¹³¹I S-factor for reference phantom, P,

A_(T) whole-body activity at time, T,

F_(T) fraction of A_(T) that is in the lungs at time, T,

λ_(LU) effective clearance rate from lungs (=ln(2)/T_(E); with T_(E)=effective half-life),

λRB effective clearance rate from remainder body (=ln(2)/T_(RB), with T_(RB)=effective half-life in remainder body),

S_(LU←TB) ^(P) total-body to lung ¹³¹I S-factor for reference phantom. P,

M_(RB) ^(P) total-body mass of reference phantom. P,

M_(LU) ^(P) lung mass of reference phantom. P.

Equation (2) describes a model in which radioiodine uptake in tumor-bearing lungs is assumed instantaneous relative to the clearance kinetics. Clearance is modeled by an exponential expression with a clearance rate constant, λ_(LU), and corresponding effective half-life, T_(E). At a particular time, T, after administration, the fraction of whole-body activity that is in the lungs is given by the parameter, F_(T). Activity that is not in the lungs (i.e., in the remainder body (RB)) is also modeled by an exponential clearance (Equation (3)), but with a different rate constant, λ_(RB). At time, T, the fraction of whole-body activity in this compartment is 1-F_(T). Equation (4) can be obtained from the following reference: Loevinger et al., MIRD Primer for Absorbed Dose Calculations, Revised Edition. The Society of Nuclear Medicine, Inc. (1991).

Using equations (2) and (3) to replace A_(LU)(t) and A_(RB)(t) in equation 1, the dose rate to lungs at time, T, for phantom, P, is:

DR^(P)(T)=A _(T) ·F _(T) ·S _(LU←LU) ^(P) +A _(T)·(1−F _(T))·S _(LU←RB) ^(P),   (5)

Derived Activity Constraint

In 115, once we have the dose-rate at a certain time (e.g., 48 h) as the relative constraint for a certain phantom, then dose-rate constraints (DRCs) at this same time (e.g., 48 h) can be derived for different reference phantoms. Thus, for example, If we assume that the dose-rate to the lungs at 48 h is the relevant constraint on avoiding prohibitive lung toxicity, then, one may derive 48-hour activity constraints for different reference phantoms that give a 48 h dose rate equal to a pre-determined fixed dose rate constraint, denoted, DRC. By re-ordering expression 5 and renaming A_(T) to A_(DRC) ^(P), the activity constraint for phantom P so that DR^(P)(48 h)=DRC, we get:

$\begin{matrix} {A_{DRC}^{P} = {\frac{DRC}{{F_{48} \cdot S_{{LU}\leftarrow{LU}}^{p}} + {\left( {1 - F_{48}} \right) \cdot S_{{LU}\leftarrow{RB}}^{P}}}.}} & (6) \end{matrix}$

In equation 6, A_(DRC) ^(P), depends upon the fraction of whole-body activity in lungs at 48 hours and also on the reference phantom that best matches the patient characteristics.

Corresponding Administered Activity

In 120, once we have the dose rate constraints for different phantoms, corresponding administered activity can be found for those phantoms. Equation (6) gives the 48 hour whole-body activity constraint so that the dose rate to lungs at 48 hours does not exceed DRC. The corresponding constraint on the maximum administered activity, AA_(max), can be derived by using equations 2 and 3, to give an expression for the total-body activity as a function of time:

$\begin{matrix} {{A_{TB}(t)} = {{\frac{A_{T} \cdot F_{T}}{^{{{- \lambda_{LU}} \cdot T}\;}}^{{- \lambda_{LU}} \cdot T}} + {\frac{A_{T} \cdot \left( {1 - F_{T}} \right)}{^{{- \lambda_{RB}} \cdot T}}{^{{- \lambda_{RB}} \cdot T}.}}}} & (7) \end{matrix}$

Replacing A_(T) with A_(DRC) ^(P) and setting t=0, the following expression is obtained for AA_(max):

$\begin{matrix} {{AA}_{{ma}\; x} = {\frac{A_{DRC}^{P} \cdot F_{48}}{^{{- \lambda_{\omega}} \cdot 48}} + {\frac{A_{DRC}^{P} \cdot \left( {1 - F_{T}} \right)}{^{{- \lambda_{RB}} \cdot 48}}.}}} & (8) \end{matrix}$

The denominator in each term of this expression scales the activity up to reflect the starting value needed to obtain A_(TB) (48 h)=A_(DRC) ^(P). AA_(max) is shown to be dependent on λ_(DU), and λ_(RB) (or equivalently on T_(E) and T_(RB)).

Mean Lung Absorbed Dose

In 125, the mean absorbed dose may be obtained by integrating the dose rate from zero to infinity. Thus, the mean lung absorbed does can be obtained by integrating equation (1) from 1 to infinity:

DLU=Ã _(LU) ·S _(LU←LU) ^(P) +Ã _(RB) ·S _(LU←RB) ^(P)   (9)

Integrating the expressions for lung and remainder body activity as a function of time, (equations (2) and (3), respectively) and replacing parameters with the 48 hour constraint values the following expressions are obtained for Ã_(LU) and Ã_(RB):

$\begin{matrix} {{{\overset{\sim}{A}}_{LU} = {\frac{A_{DRC}^{P} \cdot F_{48} \cdot ^{\frac{l\; {n{(2)}}}{T_{E}}T}}{\ln (2)} \cdot T_{E}}},} & (10) \\ {{\overset{\sim}{A}}_{RB} = {\frac{A_{DRC}^{P} \cdot \left( {1 - F_{48}} \right) \cdot ^{\; {\frac{l\; {n{(2)}}}{T_{RB}} \cdot T}}}{\ln (2)} \cdot {T_{\; B}.}}} & (11) \end{matrix}$

In equations (10) and (11), the λ values have been replaced to explicitly show the dependence of the cumulated activities on the clearance half-lives.

If T_(RB) is kept constant and T_(E) is varied, the minimum absorbed dose to the lungs will occur at a T_(E) value that gives a minimum for equation (9). This can be obtained by differentiating with respect to T_(E), setting the resulting expression to zero and solving for T_(E). This gives T_(E)=ln(2)48 h=33 h.

Electron v. Photon Contribution to the Lung Dose

Since almost all of the activity in tumor-bearing lungs would be localized to tumor cells, it is instructive to separate the electron contribution to the estimated lung dose from the photon contribution. The electron contribution would be expected, depending upon the tumor geometry (11), to irradiate tumor cells predominantly, while the photon contribution will irradiate lung parenchyma. The dose contribution from the remainder body is already limited to photon emissions. The photon only lung to lung S-value (S_(LU←LU) ^(P)) for a phantom, P, is obtained from the S-factor value and the delta value for electron emissions of ¹³¹I:

$\begin{matrix} {{{SP}_{{LU}\leftarrow{LU}}^{P} = {S_{{LU}\leftarrow{LU}}^{P} - \frac{\Delta_{electron}^{131_{I}}}{M_{LU}^{P}}}},} & (12) \end{matrix}$

Δ_(electron) ¹³¹ ¹ Total energy emitted as electrons per disintegration of ¹³¹I. Replacing SP_(LU←LU) ^(P) for S_(LU←LU) ^(P) in equation 9 gives the absorbed dose to lungs from photon emissions only.

Parameter Values

The table in FIG. 6 lists the reference phantom parameter values used in the calculations. The masses, lung-to-lung and total body-to-lung S-values listed were obtained from the OLINDA dose calculation program. The remainder body (RB)-to-lung and lung-to-lung photon-only S values were calculated using equations (4) and (12), respectively. The effective clearance half-life of radioiodine activity not localized to the lungs, T_(RB), was set to 10 or 20 h. These values correspond to reported values for whole-body clearance with or without recombinant TSH, respectively. The effective clearance half-life for activity in tumor-bearing lungs, T_(E), was varied from 20 to 100 h. The fraction of whole-body activity in the lungs at 48 h, F₄₈, was varied from 0.6 to 1.

EXAMPLE

To derive the dose-rate to lungs associated with the 80 mCi, 48 h constraint we assume that 90% of the whole-body activity is uniformly distributed in the lungs (F₄₈=0.9). The original reports describing the 80 mCi, 48 h limit do not provide a value for this parameter; the value chosen is consistent with the expected biodistribution in patients with disease that is dominated by diffuse lung metastases. As noted above, the 80 mCi activity constraint was derived primarily from results obtained in females. Accordingly, the conversion from activity to dose-rate is performed using S-factors and masses for the adult female phantom. Using equation, (5), with F₄₈=0.9, the dose-rate constraint (DRC) corresponding to the 48 h, 80 mCi limit is 41.1 cGy/h. This is the estimated dose-rate to the lungs when 80 mCi of ¹³¹I are uniformly distributed in the lungs of an adult female whose anatomy is consistent with the standard female adult phantom geometry. Implicit in the 80 mCi at 48 h constraint is that radiation induced pneumonitis and pulmonary fibrosis will be avoided as long as the dose-rate is not in excess of 41.1 cGy/h at 48 h after ¹³¹I administration. If we assume that this dose-rate based constraint applies to pediatric patients, then using equation 6, we may calculate the 48 h activity limitation if the patient anatomy is consistent with the 15 or 10 year-old standard phantom. FIG. 7 provides the 48 h whole-body activity retention values for different phantoms and F₄₈ values. Since the guidelines were developed with data from females, the 80 mCi rule applied to adult males gives a 48 h whole-body activity constraint of 3.73 GBq (101 mCi) at F₄₈=0.9; corresponding values for the 15-year old and 10-year old phantoms are 2.46 GBq (66.4 mCi) and 1.73 GBq (46.7 mCi). It is important to note that because it is a dose-rate, DRC, does not depend upon the clearance parameters. The value chosen, does, however, depend upon the assumed lung fraction of whole-body activity. FIG. 8 lists the DRC values for different F₄₈ values. All of the results presented will scale linearly by DRC value.

The majority of patients with diffuse ¹³¹I-avid lung metastases exhibit prolonged whole-body retention. In such cases, the whole-body kinetics are dominated by tumor-associated activity. Assuming that 90% of the whole-body activity is in the lungs and that this clears with an effective half-life of 100 h, while the remainder activity clears with an effective half-life of 20 h (corresponding to a treatment plant that includes hormone withdrawal), equation (8) may be used to calculate the administered activity that will yield the corresponding 48 h activity constraint for each phantom. The administered activity values are 6.64, 5.23, 4.37 and 3.08 GBq (180, 143, 118 and 83.2 mCi), respectively, for adult male, female, 15 year old and 10 year old standard phantom anatomies. These values depend on the assumed clearance half life of activity in the remainder of the body. If an effective half-life of 10 h (consistent with use of recombinant human TSH (rhTSH)) is assumed, the corresponding administered activity values are: 15.1, 12.0, 9.92 and 6.98 GBq (407, 323, 268 and 189 mCi). FIG. 9 depicts administered activity limits for different phantoms and F₄₈ values as a function of T_(E) when T_(RB) is set to 20 h. FIG. 10 depicts corresponding results when T_(RB) equals 10 h. The plots show that at T_(E) greater than three to four times T_(aRB) the administered activity limit is largely independent of lung clearance half-life but, as shown by the equidistant spacing of the curves with increasing F₄₈, remains linearly dependent on the fraction of whole-body activity that is in the lungs at 48 h. When T_(E) approaches the lower T_(RB) value as in FIG. 9, AA_(max) increases rapidly and appears to converge. This reflects the condition of a partitioned activity distribution that clears from lungs and remainder body at the same effective half-life; the AA values at T_(E)=T_(RB)=20 h are not the same because the dose-rate, which is used to determine AA_(max) will be different due to the physical distribution of the activity even when the half-lives are the same. The rapid increase in AA_(max) at lower T_(E) values reflects the need to increase patient administered activity as the clearance rate increases.

FIG. 11 depicts the absorbed dose to lungs as a function of T_(E) for different F₄₈ values. Since the administered activities are adjusted to reach a constant 48 h dose-rate in lungs, the absorbed dose curves are essentially independent of phantom geometry with less than 2% difference in lung absorbed dose vs T_(E) profiles across the 4 standard phantom geometries (data not shown). Both sets of curves show a minimum absorbed dose at T_(E)=33.3 h (=ln(2)×48 h). The minimum absorbed dose is 53.6 Gy for T_(RB)=20 h and ranges from 53.6 (F₄₈=1) to 54.2 Gy (F₄₈=0.6) for T_(RB)=10 h.

Less is known regarding the effects of lung irradiation on pediatric patients or patients with already compromised lung function. In such cases a more conservative dose-rate limit may be appropriate. As noted earlier, the results shown in FIGS. 7, and 9-11 scale linearly with DRC, the table in FIG. 12 summarizes the relevant results for the different phantoms at F₄₈=0.9 when DRC=20 cGy/h. In the adult female phantom, this corresponds to 1.44 GBq (38.9 mCi) retained in the whole body at 48 h.

All of the lung absorbed dose values shown on FIG. 11 and listed in the table in FIG. 12 are well above the reported 24 to 27 Gy MTD for adult lungs. The discrepancy may be explained by considering the photon and electron fraction of this absorbed dose. The electron emissions are deposited locally, most likely within the thyroid carcinoma cells that have invaded the lungs, while the photon contribution would irradiate the total lung volume. Using equations (9) and (12), the lung absorbed dose attributable to photons may be calculated. Absorbed dose vs T_(E) curves are depicted in FIG. 13 or the adult female phantom and for the two different remainder body effective clearance half-lives considered. The photon absorbed dose is less dependent upon clearance of lung activity (i.e., T_(E)) and more dependent on remainder body clearance. As shown in FIG. 13, there is a greater than two-fold increase in the photon absorbed dose as the clearance rate is doubled and F₄₈=0.6. The increase depends upon the spatial distribution so that at F₄₈=0.9, the dose increases by a factor of 1.6.

Unlike the total dose, the photon dose is also more heavily dependent upon the phantom. At T_(E)=100 h, the photon lung dose to the adult male ranges from 8.75 (F₄₈=1.0) to 9.03 (F₄₈=0.6) Gy when TRB=20 h; the corresponding values when T_(RB)=10 h, are 8.75 (F₄₈=1.0) and 9.56 (F₄₈=0.6) Gy. In the 15-year-old, the corresponding values are: 7.85, 8.10, 7.85 and 8.55 Gy; corresponding values for the 10-year-old are: 7.09, 7.33, 7.09 and 7.78 Gy. 3D-RD (3D-Radiobiological Dosimetry)

In one embodiment, a method is provided that incorporates radiobiological modeling to account for the spatial distribution of absorbed dose and also the effect of dose-rate on biological response. The methodology is incorporated into a software package which is referred to herein as 3D-RD (3D-Radiobiological Dosimetry). Patient-specific, 3D-image based internal dosimetry is a dosimetry methodology in which the patient's own anatomy and spatial distribution of radioactivity over time are factored into an absorbed dose calculation that provides as output the spatial distribution of absorbed dose.

FIGS. 3-4 illustrate actual images of patients. FIG. 5 illustrates the output of a 3D-RD calculation. FIG. 14 sets forth a method for a 3D-RD calculation. Referring to FIG. 14, in 1405, anatomical images of the patient are obtained. For example, one or more computed tomography (CT) images, such as the images illustrated in FIG. 15A, one or more single photon emission computed tomography (SPECT) images, such as the images illustrated in FIG. 15B, and/or one or more positron emission tomography (PET) images can be input. A CT image can be used to provide density and composition of each voxel for use in a Monte Carlo calculation. A CT image can also be used to define organs or regions of interest for computing spatially averaged doses. For example, a CT image can show how a tumor is distributed in a particular organ, such as a lung. A longitudinal series of PET or SPECT images can be used to perform a voxel-wise time integration and obtain the cumulated activity or total number of disintegrations on a per voxel basis. If multiple SPECT or PET studies are not available, a single SPECT or PET can be combined with a series of planar images. By assuming that the relative spatial distribution of activity does not change over time, it is possible to apply the kinetics obtained from longitudinal planar imaging over a tumor or normal organ volume to the single SPECT or PET image, thereby obtaining the required 3-D image of cumulated activity. The results of such a patient-specific 3-D imaging-based calculation can be represented as a 3-D parametric image of absorbed dose, as dose volume histograms over user-defined regions of interest or as the mean, and range of absorbed doses over such regions. Such patient-specific, voxel-based absorbed dose calculations can help better predict biological effect of treatment plans. The approach outlined above provides estimated total absorbed dose for each voxel, collection of voxels or region. An alternative approach that makes it easier to obtain radiobiological parameters such as the biologically effective dose (BED) is to estimate the absorbed dose for each measure time-point and then perform the voxel-wise time integration to get the total absorbed dose delivered. This approach makes it easier to calculate dose-rate parameters on a per voxel basis which are needed for the BED calculation.

In 1408, images are obtained relating to radioactive distribution across time.

FIG. 15C illustrates an example of kinetic parameter images; in this case the effective half-life for each voxel is displayed in a color-coded image. These kinetic parameter images provide an estimate of clearance rate of the radiological material in each voxel in a 3D image.

In 1410, the images related to the radioactive distribution are registered across time. At this point, one of at least two options (1415-1440 or 1450-1465) can be followed. In 1415, each radioactivity image can be combined with an anatomy image. The anatomical image voxel values can be utilized to assign density and composition (i.e., water, air and bone).

In 1420, Monte Carlo simulations are nm on each activity image. Thus, the 3-D activity image(s) and the matched anatomical image(s) can be used to perform a Monte Carlo calculation to estimate the absorbed dose at each of the activity image collection times by tallying energy deposition in each voxel.

In 1425, the dose image at each point in time is used to obtain a dose-rate image and a total dose image.

In 1430, the dose-rate and the total dose image are utilized to obtain the BED image. In 1440, the BED image is utilized to obtain the EUD of BED value for a chosen anatomical region.

In the other option, after 1410, activity images can be integrated across time voxel-wise to obtain a cumulated activity (CA) image. In 1455, the CA image is combined with an anatomy image. In 1460, a Monte Carlo (MC) calculation is run to get the total dose image. In 1465, the dose image is utilized to obtain the EUD of the dose value for a chosen anatomical region.

EUD and BED formulas (as described in more detail below and also illustrated in FIGS. 16-20) can be utilized to incorporate the radiological images into the anatomical image(s). The spatial absorbed dose distribution can be described by the equivalent uniform dose (EUD, defined on a per structure basis). The rate at which the material is delivered can be described by the biologically effective dose (BED, defined on a per voxel basis).

EUD and BED

The uniformity (or lack thereof) of absorbed dose distributions and their biological implications have been examined. Dose-volume histograms have been used to summarize the large amount of data present in 3-D distributions of absorbed dose in radionuclide dosimetry studies. The EUD model introduces the radiobiological parameters, α and β, the sensitivity per unit dose and per unit dose squared, respectively, in the linear-quadratic dose-response model. The EUD model converts the spatially varying absorbed dose distribution into an equivalent uniform absorbed dose value that would yield a biological response similar to that expected from the original dose distribution. This provides a single value that may be used to compare different dose distributions. The value also reflects the likelihood that the magnitude and spatial distribution of the absorbed dose is sufficient for tumor kill.

It is known that dose rate influences response. The BED formalism, sometimes called Extrapolated Response Dose, was developed to compare different fractionation protocols for external radiotherapy. BED may be thought of as the actual physical dose adjusted to reflect the expected biological effect if it were delivered at a reference dose-rate. As in the case of EUD, by relating effects to a reference value, this makes it possible to compare doses delivered under different conditions. In the case of EUD the reference value relates to spatial distribution and is chosen to be a uniform distribution. In the case of BED the reference value relates to dose rate and is chosen to approach zero (total dose delivered in an infinite number of infinitesimally small fractions).

As described above with respect to 1415, the patient-specific anatomical image(s) are combined with radioactivity images into paired 3D data sets. Thus, the anatomical image voxel values can be utilized to assign density and composition (i.e., water, air and bone). This information, coupled with assignment of the radiobiological parameters, α, β, μ, the radiosensitivity per unit dose, radiosensitivity per unit dose squared and the repair rate assuming an exponential repair process, respectively, is used to generate a BED value for each voxel, and subsequently an EUD value for a particular user-defined volume.

In external radiotherapy, the expression for BED is:

$\begin{matrix} {{BED} = {{{Nd}\left( {1 + \frac{d}{\alpha/\beta}} \right)}.}} & (1) \end{matrix}$

This equation applies for N fractions of an absorbed dose, d, delivered over a time interval that is negligible relative to the repair time for radiation damage (i.e., at high dose rate) where the interval between fractions is long enough to allow for complete repair of repairable damage induced by the dose d; repopulation of cells is not considered in this formulation but there are formulations that include this and this could easily be incorporated in embodiments of the present invention. The parameters, .alpha. and .beta. are the coefficients for radiation damage proportional to dose (single event is lethal) and dose squared (two events required for lethal damage), respectively.

A more general formulation of equation (1) is:

BED(T)=(D _(T)(T)·RE(T)   (2),

where BED(T) is the biologically effective dose delivered over a time T, D_(T)(T) is the total dose delivered over this time and RE(T) is the relative effectiveness per unit dose at time, T. The general expression for RE(T) assuming a time-dependent dose rate described by D(t) is given by:

$\begin{matrix} {{{RE}(T)} = {1 + {\frac{2}{{D_{T}(T)}\left( \frac{\alpha}{\beta} \right)} \times {\int_{O}^{T}{{{t} \cdot {\overset{.}{D}(t)}}{\int_{0}^{t}{{{w} \cdot {\overset{.}{D}(w)}}{^{- {\mu {({t - w})}}}.}}}}}}}} & (3) \end{matrix}$

The second integration over the time-parameter, w, represents the repair of potentially lethal damage occurring while the dose is delivered, i.e., assuming an incomplete repair model. If we assume that the dose rate for radionuclide therapy, D(t), at a given time, t, can be expressed as an exponential expression:

{dot over (D)}(t)={dot over (D)} ₀ e ^(−A1)   (4),

Where D₀ is the initial dose rate and λ is the effective clearance rate (=ln(2)/t_(e); t_(e)=effective clearance half-life of the radiopharmaceutical), then, in the limit, as T approaches infinity, the integral in equation (3) reduces to:

$\begin{matrix} {\frac{{\overset{.}{D}}_{0}^{2}}{2{\lambda \left( {\mu - \lambda} \right)}}.} & (5) \end{matrix}$

Substituting this expression and replacing D_(T)(T) with D, the total dose delivered, and using D₀=λD, which may be derived from equation (4), we get:

$\begin{matrix} {{BED} = {D + {\frac{\beta \; D^{2}}{\alpha}{\left( \frac{\ln (2)}{{\mu \cdot t_{e}} + {\ln (2)}} \right).}}}} & (6) \end{matrix}$

In this expression, the effective clearance rate, λ, is represented by ln(2)/te. The derivation can be completed as discussed in the following references: Dale et al., The Radiobiology of Conventional Radiotherapy and its Application to Radionuclide Therapy, Cancer Biother Radiopharm (2005), Volume 20, Chapter 1, pages 47-51; Dale, Use of the Linear-Quadratic Radiobiological Model for Quantifying Kidney Response in Targeted Tadiotherapy, Cancer Biother Radiopharm, Volume 19, Issue 3, pages 363-370 (2004).

In cases where the kinetics in a particular voxel are not well fitted by a single decreasing exponential alternative, formalisms have been developed that account for an increase in the radioactivity concentration followed by exponential clearance. Since the number of imaging time-points typically collected in dosimetry studies would not resolve a dual parameter model (i.e., uptake and clearance rate), in one embodiment, the methodology assumes that the total dose contributed by the rising portion of a tissue or tumor time-activity curve is a small fraction of the total absorbed dose delivered.

Equation (6) depends upon the tissue-specific intrinsic parameters, α, β and μ. These three parameters are set constant throughout a user-defined organ or tumor volume. The voxel specific parameters are the total dose in a given voxel and the effective clearance half-life assigned to the voxel. Given a voxel at coordinates (i,j,k), D^(ijk) and t_(e) ^(ijk) are the dose and effective clearance half-life for the voxel. The imaging-based formulation of expression (6) that is incorporated into 3D-RD is then:

$\begin{matrix} {{BED}^{ijk} = {D^{ijk} + {\frac{\beta \; D^{{ijk}^{2}}}{\alpha}{\left( \frac{\ln (2)}{{\mu \cdot t_{e}^{ijk}} + {\ln (2)}} \right).}}}} & (7) \end{matrix}$

The user inputs values of α, β and μ for a particular volume and D^(ijk) and t_(e) ^(ijk) are obtained directly from the 3-D dose calculation and rate image, respectively. This approach requires organ or tumor segmentation that corresponds to the different α, β and μ values. The dose values are obtained by Monte Carlo calculation as described previously, and the effective clearance half-lives are obtained by fitting the data to a single exponential function. Once a spatial distribution of BED values has been obtained a dose-volume histogram of these values can be generated. Normalizing so that the total area under the BED (differential) DVH curve is one, converts the BED DVH to a probability distribution of BED values denoted, P(.PSI.), where .PSI. takes on all possible values of BED. Then, following the derivation for EUD as described in O'Donoghue's Implications of Nonuniform Tumor Doses for Radioimmunotherapy, J Nucl Med., Volume 40, Issue 8, pages 1337-1341 (1999), the EUD (1440) is obtained as:

$\begin{matrix} {{EUD} = {{- \frac{1}{\alpha}}{{\ln \left( {\int_{0}^{\infty}{{P(\psi)}^{{- a}\; \psi}D\; \psi}} \right)}.}}} & (8) \end{matrix}$

The EUD of the absorbed dose distribution (1465), as opposed to the BED distribution (1440), can also be obtained using equation (8), but using a normalized DVH of absorbed dose values rather than BED values. Expression (8) may be derived by determining the absorbed dose required to yield a surviving fraction equal to that arising from the probability distribution of dose values (absorbed dose or BED) given by the normalized DVH.

A rigorous application of equation (7) would require estimation of the absorbed dose at each time point (as in 1420); the resulting set of absorbed dose values for each voxel would then be used to estimate t_(c) ^(ijk) (1425). In using activity-based rate images to obtain the t_(e) ^(ijk) , instead of the absorbed dose at each time point, the implicit assumption is being made that the local, voxel self-absorbed dose contribution is substantially greater than the cross-voxel contribution. This assumption avoids the need to estimate absorbed dose at multiple time-points, thereby substantially reducing the time required to perform the calculation. In another embodiment of the invention absorbed dose images at each time-point are generated and used for the BED calculation (1430), thereby avoiding the assumption regarding voxel self-dose vs cross-dose contribution.

Radiobiological Parameters for Clearance Rate Effect Example

The illustrative simplified examples and also the clinical implementation involve dose estimation to lungs and to a thyroid tumor. Values of α and β for lung, and the constant of repair, μ, for each tissue was taken from various sources. The parameter values are listed in the table in FIG. 21.

Clearance Rate Effect Example

As explained above, a sphere can be generated in a 563 matrix such that each voxel represents a volume of (0.15 cm)³. All elements with a centroid greater than 1 cm and less than or equal to (2.0 cm)⅓ from the matrix center (at 28,28,28) were given a clearance rate value (λ) corresponding to a half life of 2 hours. Those elements with a center position less than or equal to 1.0 cm from the center voxel were assigned a λ value equivalent to a 4 hour half life. In this way an outer shell (with 2 hour half life) was separated from an inner sphere (with 4 hour half life) (FIG. 23). This allowed both regions to have nearly equivalent volumes. The procedure was used to generate a matrix representing a sphere with a uniform absorbed dose distribution despite having non uniform clearance rate. This is accomplished by varying the initial activity such that the cumulative activity of both regions is identical. These two matrices were input into 3D-RD for the BED and EUD calculations. Input of a dose distribution rather than an activity distribution was necessary to make comparison with an analytical calculation possible. The partial-volume effects of a voxelized vs. idealized sphere were avoided by using the shell and sphere volumes obtained from the voxelized sphere rather than from a mathematical sphere. The impact of sphere voxelization on voxel-based MC calculations has been previously examined.

Absorbed Dose Distribution Effects. To demonstrate the impact of dose distribution on EUD, the following model was evaluated (FIG. 24). First, a uniform density sphere (1.04 g/cc in both regions) was evaluated with a uniform absorbed dose distribution of 10 Gy. Second, the uniform sphere was divided into two equal volume regions. The inner sphere was assigned zero absorbed dose while the outer shell was assigned an absorbed dose of 20 Gy. The effective half-life was 2 hours in both regions. Again the whole sphere average dose was 10 Gy.

Density Effects. To illustrate the effect of density differences, a sphere with radius 1.26 cm was created that had unit cumulated activity throughout, but a density of 2 g/cc in a central spherical region with radius 1 cm and 1 g/cc in the surrounding spherical shell (FIG. 25). The input parameters were chosen to yield a mean dose over the whole sphere of 10 Gy. Since, for a constant spatial distribution of energy deposition, the absorbed dose is a function of the density, the absorbed dose in the center is less than the absorbed dose of the shell. The distribution was selected so that the average over the two regions was 10 Gy. 3D-RD was used to generate a spatial distribution of absorbed dose values which were then used to estimate EUD over the whole sphere.

Application to a Patient Study

The 3D-RD dosimetry methodology was applied to an 11 year old female thyroid cancer patient who has been previously described in a publication on MCNP-based 3D-ID dosimetry.

Imaging. SPECT/CT images were obtained at 27, 74, and 147 hours post injection of a 37 MBq (1.0mCi) tracer 131I dose. All three SPECT/CT images focused on the chest of the patient and close attention was directed at aligning the patient identically for each image. The images were acquired with a GE Millennium VG Hawkeye system with a 1.59 cm thick crystal.

An OS-EM based reconstruction scheme was used to improve quantization of the activity map. A total of 10 iterations with 24 subsets per iteration was used. This reconstruction accounts for effects including attenuation, patient scatter, and collimator response. Collimator response includes septal penetration and scatter. The SPECT image counts were converted to units of activity by accounting for the detector efficiency and acquisition time. This quantification procedure, combined with image alignment, made it possible to follow the kinetics of each voxel. Using the CTs, which were acquired with each SPECT, each subsequent SPECT and CT image was aligned to the 27 hour 3-D image set. A voxel by voxel fit to an exponential expression was then applied to the aligned data set to obtain the clearance half-time for each voxel.

To obtain mean absorbed dose, mean BED and EUD, as well as absorbed dose and BED-volume-histograms, voxels were assigned to either tumor or normal lung parenchyma using an activity threshold of 21% of highest activity value.

Spherical Model Example

A spherical model was used to validate and illustrate the concepts of BED and EUD.

Clearance Rate Effects. Assuming that the sphere was lung tissue and applying the radiobiological parameters listed in the table of FIG. 21, the BED value in the slower clearing region, corresponding to the inner sphere with an activity clearance half-life of 4 hours, was 13.14 Gy. The faster clearing region (outer shell, 2 hr half-life) yielded a BED value of 15.69 Gy. The same model using the radiobiological values for tumor gave 10.09 Gy and 11.61 Gy for the slower clearing and faster clearing regions, respectively. The mean absorbed dose (AD) value for all these regions was 10 Gy.

Absorbed Dose Distribution Effects. The EUD value over the whole sphere when a uniform activity distribution was assumed recovered the mean absorbed dose of 10 Gy. A non-uniform absorbed dose distribution was applied such that the inner sphere was assigned an absorbed dose of zero, and an outer shell of equal volume, an absorbed dose of 20 Gy. In this case, the mean absorbed dose is 10 Gy, but the EUD was 1.83 Gy. The substantially lower EUD value is no longer a quantity that may be obtained strictly on physics principles, but rather is dependent on the applied biological model. The true absorbed dose has been adjusted to reflect the negligible probability of sterilizing all cells in a tumor volume when half of the tumor volume receives an absorbed dose of zero.

Density Effects. In the sphere with non-uniform density (inner sphere density of 2 g/cc, outer shell of equal volume (1 g/cc)) and an average absorbed dose of 10 Gy, the EUD over the whole sphere was 6.83 Gy. The EUD value is lower than the absorbed dose value to reflect the dose non-uniformity in spatial absorbed dose (inner sphere=5 Gy, outer shell=15 Gy) arising from the density differences.

Application to a Patient Study. A 3D-RD calculation was performed for the clinical case described in the methods. A dosimetric analysis for this patient, without the radiobiological modeling described in this work has been previously published using the Monte Carlo code MCNP as opposed to EGSnrc which was used in this work. The clinical example illustrates all of the elements investigated using the simple spherical geometry. As shown on the CT scan (FIG. 26a ), there is a highly variable density distribution in the lungs due to the tumor infiltration of normal lung parenchyma. Coupled with the low lung density, this gives a density and tissue composition that includes air, lung parenchyma and tumor (which was modeled as soft tissue). As shown on FIGS. 26b and 26c , the activity and clearance kinetics of 131I are also variable over the lung volume. These two data sets were used to calculate the cumulated activity images shown in FIG. 26 d.

A comparison between the EGS-based 3D-RD calculation and the previously published MCNP-based calculation was performed. FIG. 27 depicts the DVH of the absorbed dose distribution obtained with 3D-RD superimposed on the same plot as the previously published DVH. Good overall agreement between the two DVHs is observed and the mean absorbed doses, expressed as absorbed dose per unit cumulated activity in the lung volume are in good agreement, 3.01.times.10-5 and 2.88.times.10-5 mGy/MBq-s per voxel, for the published, MCNP-based, and 3D-RD values, respectively.

FIGS. 28 and 29 depict the results obtained with the radiobiological modeling capabilities of 3D-RD. FIG. 28 depicts a parametric image of BED values. Within this image the spotty areas of highest dose are areas where high activity and low density overlap. In graph A of FIG. 29, normalized (so that the area under the curve is equal to 1) DVH and BED DVH (BVH) are shown for tumor voxels. The near superimposition of DVH and BVH suggests that dose rate will have a minimal impact on tumor response in this case. Graph B of FIG. 29 depicts the normalized BVH for normal lung parenchyma. The DVH and BVH are given in Gy and reflect the predicted doses resulting from the administered therapeutic activity of 1.32 GBq (35.6 mCi) of 131I. These plots may be used to derive EUD values. It is important to note that the volume histograms must reflect the actual absorbed dose delivered and not the dose per unit administered activity. This is because the EUD is a nonlinear function of absorbed dose. The model relies on estimation of a tumor control probability to yield the equivalent uniform dose. If the data used to estimate EUD are expressed as dose per administered activity the EUD value will be incorrect. Mean absorbed dose, mean BED, and EUD are summarized in table 2. The EUD value for tumor, which accounts for the effect of a non-uniform dose distribution, was approximately 43% of the mean absorbed dose. This reduction brings the absorbed dose to a range that is not likely to lead to a complete response. The analysis demonstrates the impact of dose non-uniformity on the potential efficacy of a treatment.

CONCLUSION

While various embodiments have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art(s) that various changes in form and detail can be made therein without departing from the spirit and scope. In fact, after reading the above description, it will be apparent to one skilled in the relevant art(s) how to implement alternative embodiments. Thus, the present embodiments should not be limited by any of the above described exemplary embodiments.

In addition, it should be understood that any figures which highlight the functionality and advantages, are presented for example purposes only. The disclosed architecture is sufficiently flexible and configurable, such that it may be utilized in ways other than that shown. For example, the steps listed in any flowchart may be re-ordered or only optionally used in some embodiments.

Further, the purpose of the Abstract of the Disclosure is to enable the U.S. Patent and Trademark Office and the public generally, and especially the scientists, engineers and practitioners in the art who are not familiar with patent or legal terms or phraseology, to determine quickly from a cursory inspection the nature and essence of the technical disclosure of the application. The Abstract of the Disclosure is not intended to be limiting as to the scope in any way.

Finally, it is the applicant's intent that only claims that include the express language “means for” or “step for” be interpreted under 35 U.S.C. 112, paragraph 6. Claims that do not expressly include the phrase “means for” or “step for” are not to be interpreted under 35 U.S.C. 112, paragraph 6. 

What is claimed is:
 1. A computerized method for determining an optimum amount of radioactivity to administer to a patient, comprising: assuming an activity retention limit; utilizing the activity retention limit to determine a dose rate for a phantom category; utilizing the dose rate for the phantom category to determine the dose rate for a second phantom category; and utilizing the dose rate for a second phantom category to find information regarding the second phantom category.
 2. The method of claim 1, wherein the information is the activity retention limit for the second phantom category.
 3. The method of claim 1, further comprising: obtaining the mean absorbed dose by integrating the dose rate over time; using the mean absorbed dose to determine an optimum amount of radioactivity to administer to the patient.
 4. The method of claim 1, wherein a dose rate resulting from short range particulate emissions is distinguished from a dose rate resulting from longer range emissions.
 5. The method of claim 4, wherein the short range particulate emissions are electrons.
 6. The method of claim 4, wherein the longer range emissions are photons.
 7. The method of claim 1, wherein utilizing the activity retention limit to determine the dose rate for the phantom category further comprises calculating how the dose rate is related to the activity retention limit in a particular point in time.
 8. The method of claim 1, wherein utilizing the activity retention limit to determine the dose rate for the phantom category further comprises using the following formula: DR^(P)(t)=A _(LU)(t)·S _(LU←LU) ^(P) +A _(RB)(t)·S _(LU←RB) ^(P)   (1), with: $\begin{matrix} {{{A_{LU}(t)} = {\frac{A_{T} \cdot F_{T}}{^{{{- \pi_{LU}} \cdot T}\;}}^{{- \lambda_{LU}} \cdot T}}},} & (2) \\ {{{A_{RB}(t)} = {\frac{A_{T} \cdot \left( {1 - F_{T}} \right)}{^{{- \lambda_{RB}} \cdot T}}^{{- \lambda_{RB}} \cdot T}}},} & (3) \\ {{S_{{LU}\leftarrow{RB}}^{P} = {{S_{{LU}\leftarrow{TB}}^{P} \cdot \frac{M_{TB}^{P}}{M_{TB}^{P} - M_{LU}^{P}}} - {S_{{LU}\leftarrow{LU}}^{P} \cdot \frac{M_{LU}^{P}}{M_{TB}^{P} - M_{LU}^{P}}}}},} & (4) \end{matrix}$ A_(LU)(t) lung activity at time t,| S_(LU←LU) ^(P) lung to lung ¹³¹I S-factor for reference phantom, P, A_(RB)(t) remainder body activity (total-body-lung) at time, t, S_(LU←RB) ^(P) remainder body to hung ¹³¹I S-factor for reference phantom, P, A_(T) whole-body activity at time, T, F_(T) fraction of A_(T) that is in the lungs at time, T, λ_(LU) effective clearance rate from lungs (=ln(2)/T_(E); with T_(E)=effective half-life), λRB effective clearance rate from remainder body (=ln(2)/T_(RB), with T_(RB)=effective half-life in remainder body), S_(LU←TB) ^(P) total-body to lung ¹³¹I S-factor for reference phantom. P, M_(RB) ^(P) total-body mass of reference phantom. P, M_(LU) ^(P) lung mass of reference phantom. P.
 9. The method of claim 3, wherein obtaining the mean absorbed dose by integrating the dose rate over time further comprises utilizing the following formula: DLU=Ã _(LU) ·S _(LU←LU) ^(P) +Ã _(RB) ·S _(LU←RB) ^(P)   (9) with $\begin{matrix} {{{\overset{\sim}{A}}_{LU} = {\frac{A_{DRC}^{P} \cdot F_{48} \cdot ^{\frac{l\; {n{(2)}}}{T_{E}}T}}{\ln (2)} \cdot T_{E}}},} & (10) \\ {{\overset{\sim}{A}}_{RB} = {\frac{A_{DRC}^{P} \cdot \left( {1 - F_{48}} \right) \cdot ^{\; {\frac{l\; {n{(2)}}}{T_{RB}} \cdot T}}}{\ln (2)} \cdot {T_{RB}.}}} & (11) \end{matrix}$
 10. A computerized system for determining an optimum amount of radioactivity to administer to a patient, comprising: a server coupled to a network; a user terminal coupled to the network; an application coupled to the server and/or the user terminal, wherein the application is configured for: assuming an activity retention limit; utilizing the activity retention limit to determine a dose rate for a phantom category; utilizing the dose rate for the phantom category to determine the dose rate for a second phantom category; and utilizing the dose rate for a second phantom category to find information regarding the second phantom category.
 11. The system of claim 10, wherein the information is the activity retention limit for the second phantom category.
 12. The system of claim 10, wherein the application is further configured for: obtaining the mean absorbed dose by integrating the dose rate over time; using the mean absorbed dose to determine an optimum amount of radioactivity to administer to the patient.
 13. The system of claim 10, wherein a dose rate resulting from short range particulate emissions is distinguished from a dose rate resulting from longer range emissions.
 14. The system of claim 13, wherein the short range particulate emissions are electrons.
 15. The system of claim 13, wherein the longer range emissions are photons.
 16. The system of claim 10, wherein utilizing the activity retention limit to determine the dose rate for the phantom category further comprises calculating how the dose rate is related to the activity retention limit in a particular point in time.
 17. The system of claim 10, wherein utilizing the activity retention limit to determine the dose rate for the phantom category further comprises using the following formula: DR^(P)(t)=A _(LU)(t)·S _(LU←LU) ^(P) +A _(RB)(t)·S _(LU←RB) ^(P)   (1), with: $\begin{matrix} {{{A_{LU}(t)} = {\frac{A_{T} \cdot F_{T}}{^{{{- \pi_{RB}} \cdot T}\;}}^{{- \lambda_{LU}} \cdot T}}},} & (2) \\ {{{A_{RB}(t)} = {\frac{A_{T} \cdot \left( {1 - F_{T}} \right)}{^{{- \lambda_{RB}} \cdot T}}^{{- \lambda_{RB}} \cdot T}}},} & (3) \\ {{S_{{LU}\leftarrow{RB}}^{P} = {{S_{{LU}\leftarrow{TB}}^{P} \cdot \frac{M_{TB}^{P}}{M_{TB}^{P} - M_{LU}^{P}}} - {S_{{LU}\leftarrow{LU}}^{P} \cdot \frac{M_{LU}^{P}}{M_{TB}^{P} - M_{LU}^{P}}}}},} & (4) \end{matrix}$ A_(LU)(t) lung activity at time t,| S_(LU←LU) ^(P) lung to lung ¹³¹I S-factor for reference phantom, P, A_(RB)(t) remainder body activity (total-body-lung) at time, t, S_(LU←RB) ^(P) remainder body to hung ¹³¹I S-factor for reference phantom, P, A_(T) whole-body activity at time, T, F_(T) fraction of A_(T) that is in the lungs at time, T, λ_(LU) effective clearance rate from lungs (=ln(2)/T_(E); with T_(E)=effective half-life), λRB effective clearance rate from remainder body (=ln(2)/T_(RB), with T_(RB)=effective half-life in remainder body), S_(LU←RB) ^(P) total-body to lung ¹³¹I S-factor for reference phantom. P, M_(RB) ^(P) total-body mass of reference phantom. P, M_(LU) ^(P) lung mass of reference phantom. P.
 18. The method of claim 12, wherein obtaining the mean absorbed dose by integrating the dose rate over time further comprises utilizing the following formula: DLU=Ã _(LU) ·S _(LU←LU) ^(P) +Ã _(RB) ·S _(LU←RB) ^(P)   (9) with $\begin{matrix} {{{\overset{\sim}{A}}_{LU} = {\frac{A_{DRC}^{P} \cdot F_{48} \cdot ^{\frac{l\; {n{(2)}}}{T_{E}}T}}{\ln (2)} \cdot T_{E}}},} & (10) \\ {{\overset{\sim}{A}}_{RB} = {\frac{A_{DRC}^{P} \cdot \left( {1 - F_{48}} \right) \cdot ^{\; {\frac{l\; {n{(2)}}}{T_{RB}} \cdot T}}}{\ln (2)} \cdot {T_{RB}.}}} & (11) \end{matrix}$ 